I've got a list of geographical locations in an Excel spreadsheet. I want users to be able to search for a user-defined location, and then have the X number of closest locations from the list pulled up and displayed, sorted by nearest to farthest by travel time via car. I know Excel alone obviously can't do this.

I have minimal/no programming knowledge. Just an Excel tinkerer. I figure that, at a minimum, this would require utilization of a mapping service like Google Maps or Mapquest.

You say you are a "Excel tinkerer". "Mapping" in Excel is possible to some extent. I once saw a very involved Excel dashboard application that used a line chart to plot out boundaries of a study area and then VBA to determine what portion of the study area a given point was located within - all based on X\Y locations of both the study are line vertices and point locations. It was pretty impressive. Also, these articles might be of interest:

I don't necessarily condone or encourage these as solutions (as there are much better true GIS ones such as mrg's answer to this question), but like I said earlier, you said you were a Excel tinkerer with no-programming knowledge - which begs the question: What's your GIS knowledge level?